Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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matrix inverse and limit

I would like to get a better understanding of limits and matrix inverses, specifically the relationship between: $\lim_{k\rightarrow \infty}(\mathbf{A}^{-1})$ and $(\lim_{k\rightarrow ...
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1answer
18 views

Finding the marginal density function of Y

Okay, the question is like this: $f_{x}(x) = xe^{-x^2/2}$ for all $x>0$ and $Y = \ln X$, find the density of $Y$. I don't understand a particular step of this problem. First they start for $x ...
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2answers
38 views

Finding the inverse of a matrix by Gaussian elimination

I spent last hours trying to figure out how to solve the inverse matrix to this matrix: $$\begin{pmatrix} 2 &-3 & 1 \\ 1 & 2 &-1 \\ 2 & 1 & 1 \end{pmatrix}$$ The correct ...
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26 views

determinant and trace of a huge positive definite matrix

I have a problem to compute the determinant and the trace of inverse matrix: $det(\Gamma^{-1}+I_n⊗\Phi^T\Phi)$ and $tr[(\Gamma^{-1}+I_n⊗\Phi^T\Phi)^{-1}]$ where $\Gamma$ is a huge positive definite ...
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39 views

Continuity of the inverse matrix function

For a differentiation module I am taking one of the exercises (not homework) asks: Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $det(A) \neq 0$ is open. Let $A^{-1}$ be the ...
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1answer
27 views

local invertibility does not imply global invertibility

What is an example of a smooth function with continuous derivatives, that is locally invertible but not globally, and the reason for that is not injectivity. My first idea was $f:\mathbb{R}^{2}\to ...
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4answers
43 views

Set of all matrices with determinant 0, non-zero

I was assigned this problem in class: Let $f: M(n, \mathbb R) \rightarrow \mathbb R $ be given by $f(X) = det(X)$. Identify the sets $f^{-1}(0)$ and $f^{-1}(\mathbb R^*)$, where $\mathbb R^*$ denotes ...
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1answer
53 views

When inverse functions are helpful?

I pass some colloquiums to find inverse functions. But still can't understand the real help of them. Only one real world example come to my mind: converting units of measurement (but those convertions ...
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1answer
60 views

Upper bound on the inverse of a Grammian matrix

I have been trying to find a reasonable upper bound on the following: Given $n\in N$ and the Grammian matrix $A_n$ = (($f(i)$ , $f(j)$)) , $f(\lambda) = e^{\lambda t}$ for $0\le t \le 1$ and ...
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2answers
68 views

Dual quaternion inverse

Is it true that for every dual quaternion $Q$ I can find it's inverse such that $QQ^{-1} = 1?$ Using the usual definition $Q^{-1}=\frac{Q^{*}}{||Q||^2}$ doesn't work for me, since the dual part ...
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2answers
76 views

Inverse of a sum of positive definite matrices

Let $A,B$ be symmetric positive definite matrices. Let $A^{-1} = LL^T$ (Cholesky decomposition, $L$ is lower-triangular). I think the following identities are true, but I haven't found them online: $$ ...
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1answer
35 views

I need to evaluate the Inverse Trig Integral

The integral is $$\int \frac{x\,dx}{(3+2x+2x^2)}.$$ I'm stuck with breaking the denominator into $u^2+a^2$.
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75 views

inverse of Vandermonde's Matrix without using determinants

I want to show, that the Vandermonde's Matrix ...
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2answers
42 views

Inverse functions problem

There are two functions $f\colon\mathbb Q \to \mathbb Q \setminus \{-1\}$ and $g\colon\mathbb Q \to \mathbb Q \setminus \{1\}$. $$g(x) = \frac{f(x)}{f(x)+1}.$$ Prove that if there is a inverse ...
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1answer
26 views

Inverse image of an element in co-domain but not in range?

Sorry, quite new to this. I have a question that contains the image below of $g:X\rightarrow Y$ and it is asking for the inverse image of $u$. Am I correct in thinking that the answer is $\emptyset$? ...
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1answer
36 views

What is the formula for this pattern?

I'm trying to find some sort of inverse relationship between two variables. I am working on some parallax scrolling in my iPhone app and I'd like to see if someone can help me find the formula for ...
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4answers
78 views

Finding the inverse of $f(x)=|x|-2$

How would I find the inverse of the function $f(x)=|x|-2$? I have swapped $x$ and $y$, and tried to isolate $y$, reaching up to $x+2=|y|$ Whenever I see absolute values, I always break the problem up ...
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1answer
49 views

Generalized inverse of the cdf applied to a random variable equals the random variable itself almost surely?

first of all I apologize for the awful title but I really did not know how to formulate a precise question. Consider the following setup. Let $F$ be the distribution function of a random variable ...
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17 views

Integral of inverse function

On wikipedia and on the following mathstackexchange page, a formula for the sum of the integrals of a function and its inverse (with "corresponding" limits) is given, do you have a proper proof for ...
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1answer
83 views

Minimum Argument Difference to Make the Lower Bound > the Upper Bound

Assume $g$ is a function that grows asymptotically as $$ g(n) \in\frac n {log(n)} + O(\sqrt n),\,n \in \Bbb N\tag1 $$ I wish to find $h(n)$ such that $$ g(n) \le g(n+h(n)). $$ i.e. Given the bounds ...
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1answer
56 views

Looking for a commutative ring satisfying certain conditions

I'm looking for a commutative ring $R$ (with unit) which is of characteristic 2 and which possesses elements $x$ and $y$ such that the following holds $x^2$ and $y^2$ are inverses of one another but ...
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2answers
42 views

Explicit formula for inverse of upper triangular matrix inverse

I have $n \times n$ upper triangular matrix $A$ such as $$ \begin{bmatrix} x_1 & x_2 & \ldots & x_n \\ 0 & x_1 & \ldots & x_{n-1} \\ \vdots & \vdots & ...
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41 views

Inverse Laplace Transform of $ \left(\frac{1-s^{1/2}}{s^2}\right)^2$

I found this question in my N.P Bali's Engineering Mathematics 7th Edition. I could not find any solved questions related to this. How can I find the Inverse Laplace Transform of : ...
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1answer
49 views

How to calculate the inverse of the line integeral.

Let $f$ be a polynomial function, $$ f(x) = a_0 + a_1 x + ... + a_d x^d $$ where $a_0$, $a_1$, ..., $a_d$ are parameters and usually $d \le 6$. Let $g$ be the line integral of $f$, $$ g(x) = ...
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1answer
71 views

Double Think about Numerosity

According to standard mathematics, the Natural Numbers are given. Moreover, they are given as a (completed) Infinite Set. This set is commonly denoted as: $$ \mathbb{N} = \left\{ ...
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2answers
27 views

Inverse function (basic algbra math)

Consider the following function: $f(x) = {1 / (x-6) }$ Find a formula for the inverse of the function. Here is what have so far? $y = 1/(x-6)$ ---> $ x = 1/(y-6) $ But my embarrassing problem is ...
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1answer
44 views

Inverse function of $f(t)=5 +\frac{75}{1 + e^{-((t-50)/10)}}$

i need to find the inverse function of $$ v= f(t)=5 + \frac{75}{1 + e^{-\frac{t-50}{10}}} $$ so far i have $$ v - 5 = \frac{75}{1 + e^{-\frac{t-50}{10}}} $$ $$ (v-5) \left(1 + ...
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24 views

Show: $f\colon X\to Y$ bijective $\Longleftrightarrow$ f has an inverse function

As the title says, I would like to prove that $f\colon X\to Y$ bijective $\Longleftrightarrow$ f has an inverse function. Proof $\Rightarrow$ Let $f$ be bijective. That means $\forall y\in ...
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2answers
34 views

PID question in Ireland and Rosen

Context: In Ireland and Rosen's 'A classic introduction to number theory' on page 11, the proof that in a PID$=R$, there is an integer $n$ such that, for a prime $p$ and any $b\in R$, $p^n \mid b , ...
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3answers
58 views

Algebra question: Finding inverse function

This question is about finding the inverse function of $f(x)=-\sqrt{9-x^2}$ I seem to be making an error with one of the manipulations. Here is my attempt. $$x=-\sqrt{9-y^2}$$ ...
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57 views

Find the inverse of $f(x) = (x+1)/(x-8)$

Find the inverse of this function: I have gotten this far: $x = y+1/y-8$ $x(y-8) = y+1$ $x(y-8)-1=y$ $xy-8x - 1 = y$ I think I went backwards?
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5answers
96 views

How to find the inverse of $f(x) = \frac{x+2}x$?

What approach would be ideal in finding the inverse of $f(x) = \frac{x+2}x$?
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40 views

The inverse of a transpose matrix to “cancel” the transpose?

When it comes to solving and equation containing matrices I don't always understand some of the rules involved. In particular, I am trying to figure out the derivation of the Gauss-Newton algorithm. ...
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40 views

Inverse functions determination by integral

From "Inverse functions and differentiation": Integrating this relationship gives $$ f^{-1}(x)=\int\frac{1}{f'(f^{-1}(x))}\,dx + c. $$ This is only useful if the integral exists. ...
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20 views

Possible to find inverse or eigenvalues of a block diagonal matrix with upper and lower diagonal matrices

I just encountered a matrix problem of finding inverse of eigenvalues of a block diagonal matrix with upper and lower also matrices of the form where A and B are full rank matrices. Is there any ...
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2answers
45 views

prove function is surjective /analysis proofs!!

Suppose $f:(a,b)\longrightarrow\mathbb R$, differentiable, where $(a,b)\subseteq\mathbb R$ is an open interval. Assume that $f'(x)$ is not $=0$. Show that there is an open interval ...
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1answer
41 views

solve $x+\sin(x)=k$ for $x$ [duplicate]

This question has been proposed to me and thus far it has baffled me: $$ x + \sin(x) = k$$ solve for x. Another way of looking at it is find $f^{-1}(x)$ given that $f(x)=x + \sin(x)$. Wolfram alpha ...
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2answers
37 views

Calculate the product ST, and infer from it the inverse of T.

S=\begin{pmatrix} 1/2 & 1/2 & 0\\ 1 & 0 & 0\\ -3/2 & 0 & 1/2 \end{pmatrix} T= \begin{pmatrix} 0 & 1 & 0\\ 2 & -1 & 4\\ 0 & 3 & 2 \end{pmatrix} I ...
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2answers
60 views

Use Euclid's Algorithm to find the multiplicative inverse

Use Euclid's Algorithm to find the multiplicative inverse of $13$ in $\mathbf{Z}_{35}$ Can someone talk me through the steps how to do this? I am really lost on this one. Thanks
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1answer
26 views

relationships of symmetric matrices

I came across the following relationships, but I have no idea how to prove them. I would love to know they can be proved. Suppose $X$ and $Y$ are both symmetric matrices, relationship: $$(X + ...
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1answer
19 views

Class of the inverse function

The exercise goes like this: Let $f$ be an invertible function of class $C^k([a,b])$, prove that $f^{-1}$ is of the same class. But wait a second: $f(x) = x^3$ is invertible and of class $C^{\infty}$ ...
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2answers
45 views

Is $\sec^{-1}(\sec(\pi/2)) = \pi/2$?

I think it shouldn't be defined as $\pi/2$ is not in the range of the function $\sec^{-1}(x)$ Wolfram confused me by giving the answer as $\pi/2$ : Link But it mentions on another page that $\pi/2$ ...
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1answer
23 views

Trig and Inverse Trig Function Compositions

Sorry if this is a dumb question, but I honestly tried searching and all I could find was obvious stuff like $\sin(\arcsin(x)) = x$ So what is the logic behind simplifying expressions like this, ...
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1answer
34 views

Is every invertible matrix over an algebraically closed field diagonalisable?

In $\Bbb{R}$ the only invertible matrices (I can think of) that are not diagonalisable are those which stand for a rotation, but in $\Bbb{C}$ this shouldn't be a problem anymore, since rotations can ...
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1answer
64 views

Inverse Function Differential Equation [duplicate]

For the differential equation $$\frac{d}{dx}[y(x)]=y^{(-1)}(x)$$ where $y^{(-1)}(x)$ is the inverse of $y(x)$, find y(x). I gave up on finding the solution analytically pretty quickly and decided ...
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1answer
43 views

How to find the inverse of this particular symmetric matrix

Basically, I have a $n \times n$ symmetric matrix, which looks like this: $$ \begin{bmatrix} 1 & \alpha & \cdots & \alpha \\ \alpha & 1 & \cdots &\alpha \\ \vdots &\vdots ...
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1answer
34 views
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30 views

How to find the inverse system of a given one

what is the inverse formula of y[n]=x[n]*x[n+1] ? And how can I find the inverse formula/system of a given one in general? I'm having some troubles with this when it comes to some formulas.
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1answer
30 views

How do I show a left inverse of a bounded linear operator on Banach space?

If $A$ is a bounded linear operator on a Banach space X, with a left inverse $A_l^{-1}$, and P is a projection (also on X), how do I show that $A_l^{-1}P$ is also a left inverse of A (i.e. ...
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1answer
77 views

What are some practical uses of functions? [closed]

Functions are basically formal equations that relate a set of inputs to output. What are some practical uses for functions and inverse functions?